Abstract

Absorption of electromagnetic radiation by absorptive dielectric spheres such as snow grains in the near-infrared part of the solar spectrum cannot be neglected when radiative properties of snow are computed. Thus a new, to our knowledge, geometrical-optics code is developed to compute scattering and absorption cross sections of large dielectric particles of arbitrary complex refractive index. The number of internal reflections and transmissions are truncated on the basis of the ratio of the irradiance incident at the nth interface to the irradiance incident at the first interface for a specific optical ray. Thus the truncation number is a function of the angle of incidence. Phase functions for both near- and far-field absorption and scattering of electromagnetic radiation are calculated directly at any desired scattering angle by using a hybrid algorithm based on the bisection and Newton-Raphson methods. With these methods a large sphere’s absorption and scattering properties of light can be calculated for any wavelength from the ultraviolet to the microwave regions. Assuming that large snow meltclusters (1-cm order), observed ubiquitously in the snow cover during summer, can be characterized as spheres, one may compute absorption and scattering efficiencies and the scattering phase function on the basis of this geometrical-optics method. A geometrical-optics method for sphere (GOMsphere) code is developed and tested against Wiscombe’s Mie scattering code (MIE0) and a Monte Carlo code for a range of size parameters. GOMsphere can be combined with MIE0 to calculate the single-scattering properties of dielectric spheres of any size.

© 2003 Optical Society of America

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  2. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  3. Y.-X. Hu, K. Stamnes, “An accurate parameterization of the radiative properties of water clouds suitable for use in climate models,” J. Clim. 6, 728–742 (1993).
    [CrossRef]
  4. K.-N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climate implications,” Atmos. Res. 31, 271–298 (1994).
    [CrossRef]
  5. M. I. Mishchenko, “Calculation of the amplitude matrix for a nonspherical particle in a fixed orientation,” Appl. Opt. 39, 1026–1031 (2000).
    [CrossRef]
  6. F. M. Schulz, K. Stamnes, J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
    [CrossRef]
  7. H. A. Eide, J. J. Stamnes, K. Stamnes, F. M. Schulz, “New method for computing expansion coefficients for spheroidal functions,” J. Quant. Spectrosc. Radiat. Transfer 63, 191–203 (1999).
    [CrossRef]
  8. P. Yang, K. N. Liou, “Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35, 6568–6584 (1996).
    [CrossRef] [PubMed]
  9. S. C. Colbeck, “Grain clusters in wet snow,” J. Colloid Interface Sci. 72, 371–384 (1979).
    [CrossRef]
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  13. M. Sturm, K. Morris, R. Massom, “The winter snow cover of the West Antarctic pack ice: its spatial and temporal variability,” in Antarctic Sea Ice: Physical Processes, Interactions and Variability, M. O. Jeffries, ed., Vol. 74 of Antarctic Research Series (American Geophysical Union, Washington, D.C., 1998), pp. 1–18.
  14. R. A. Massom, V. I. Lytle, A. P. Worby, I. Allison, “Winter snow cover variability on East Antarctic sea ice,” J. Geophys. Res. 103, 24837–24855 (1998).
    [CrossRef]
  15. C. Haas, “The seasonal cycle of ERS scatterometer signatures over perennial Antarctic sea ice and associated surface ice properties and processes,” Ann. Glaciol. 33, 69–73 (2001).
    [CrossRef]
  16. K. Morris, M. O. Jeffries, “Seasonal contrasts in snow cover characteristics on Ross Sea ice floes,” Ann. Glaciol. 33, 61–68 (2001).
    [CrossRef]
  17. R. A. Massom, H. Eicken, C. Haas, M. O. Jeffries, M. R. Drinkwater, M. Sturm, A. P. Worby, X. Wu, V. I. Lytle, S. Ushio, K. Morris, P. A. Reid, S. G. Warren, I. Allison, “Snow on Antarctic sea ice,” Rev. Geophys. 39, 413–445 (2001).
    [CrossRef]
  18. R. A. Massom, M. R. Drinkwater, C. Haas, “Winter snow cover on sea ice in the Weddell Sea,” J. Geophys. Res. 102, 1101–1117 (1997).
    [CrossRef]
  19. X. Zhou, “Optical remote sensing of snow on sea ice: ground measurements, satellite data analysis, and radiative transfer modeling,” (Ph.D. thesis, University of Alaska, Fairbanks, Alaska, 2002).
  20. A. W. Nolin, J. Dozier, “A hyperspectral method for remotely sensing the grain size of snow,” Remote Sens. Environ. 74, 207–216 (2000).
    [CrossRef]
  21. A. A. Kokhanovsky, T. Y. Nakajima, “The dependence of phase functions of large transparent particles on their refractive index and shape,” J. Phys. D 31, 1329–1335 (1998).
    [CrossRef]
  22. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, London, 1992).
    [CrossRef]
  23. T. Y. Nakajima, T. Nakajima, A. A. Kokhanovsky, “Radiative transfer through light-scattering media with nonspherical large particles: direct and inverse problems,” in Satellite Remote Sensing of Clouds and the Atmosphere II, J. D. Haigh, ed., Proc. SPIE3220, 2–12 (1998).
    [CrossRef]
  24. D. J. Wielaard, M. I. Mishchenko, A. Macke, B. E. Carlson, “Improved T-matrix computations for large, nonabsorbing and weakly absorbing nonspherical particles and comparison with geometrical-optics approximation,” Appl. Opt. 36, 4305–4313 (1997).
    [CrossRef] [PubMed]
  25. S. G. Warren, “Optical constants of ice from the ultraviolet to the microwave,” Appl. Opt. 23, 1206–1225 (1984). See Ref. 26 for the upgrade.
  26. The updated compilation of the ice optical constant was done by B. Gao, W. Wiscombe, S. Warren and is available by anonymous ftp to climate.gsfc.nasa.gov in the directory/pub/Wiscombe/Refrac_Index/ICE.
  27. L. Kou, D. Labrie, P. Chylek, “Refractive indices of water and ice in the 0.65- to 2.5-µm spectral range,” Appl. Opt. 32, 3531–3540 (1993).
    [CrossRef] [PubMed]
  28. D. K. Perovich, J. W. Govoni, “Absorption coefficients of ice from 250 to 400 nm,” Geophys. Res. Lett. 18, 1233–1235 (1991).
    [CrossRef]
  29. C. F. Bohren, B. R. Barkstrom, “Theory of the optical properties of snow,” J. Geophys. Res. 79, 4527–4535 (1974).
    [CrossRef]
  30. E. Hecht, Optics, 3rd ed. (Addison-Wesley Longman, Reading, Mass., 1998).
  31. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).
  32. K. N. Liou, J. E. Hansen, “Intensity and polarization for single scattering by polydisperse spheres: a comparison of ray optics and Mie theory,” J. Atmos. Sci. 28, 995–1004 (1971).
    [CrossRef]
  33. W. J. Glantschnig, S.-H. Chen, “Light scattering from water droplets in the geometrical optics approximation,” Appl. Opt. 20, 2499–2509 (1981).
    [CrossRef] [PubMed]
  34. A. A. Kokhanovsky, Optics of Light Scattering Media: Problems and Solutions (Praxis, Chichester, UK, 1999).
  35. G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, Cambridge, UK, 1999).
    [CrossRef]
  36. P. J. Davis, P. Rabinowitz, Methods of Numerical Integration, 2nd ed. (Academic, Orlando, Fla., 1984), pp. 481–483.
  37. H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
    [CrossRef]
  38. E. P. Zege, A. A. Kokhanovsky, “Integral characteristics of light scattering by large spherical particles,” Izv. Atmos. Ocean. Phys. 24, 508–512 (1988).
  39. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 350–354.

2001 (3)

C. Haas, “The seasonal cycle of ERS scatterometer signatures over perennial Antarctic sea ice and associated surface ice properties and processes,” Ann. Glaciol. 33, 69–73 (2001).
[CrossRef]

K. Morris, M. O. Jeffries, “Seasonal contrasts in snow cover characteristics on Ross Sea ice floes,” Ann. Glaciol. 33, 61–68 (2001).
[CrossRef]

R. A. Massom, H. Eicken, C. Haas, M. O. Jeffries, M. R. Drinkwater, M. Sturm, A. P. Worby, X. Wu, V. I. Lytle, S. Ushio, K. Morris, P. A. Reid, S. G. Warren, I. Allison, “Snow on Antarctic sea ice,” Rev. Geophys. 39, 413–445 (2001).
[CrossRef]

2000 (2)

A. W. Nolin, J. Dozier, “A hyperspectral method for remotely sensing the grain size of snow,” Remote Sens. Environ. 74, 207–216 (2000).
[CrossRef]

M. I. Mishchenko, “Calculation of the amplitude matrix for a nonspherical particle in a fixed orientation,” Appl. Opt. 39, 1026–1031 (2000).
[CrossRef]

1999 (1)

H. A. Eide, J. J. Stamnes, K. Stamnes, F. M. Schulz, “New method for computing expansion coefficients for spheroidal functions,” J. Quant. Spectrosc. Radiat. Transfer 63, 191–203 (1999).
[CrossRef]

1998 (3)

A. A. Kokhanovsky, T. Y. Nakajima, “The dependence of phase functions of large transparent particles on their refractive index and shape,” J. Phys. D 31, 1329–1335 (1998).
[CrossRef]

R. A. Massom, V. I. Lytle, A. P. Worby, I. Allison, “Winter snow cover variability on East Antarctic sea ice,” J. Geophys. Res. 103, 24837–24855 (1998).
[CrossRef]

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
[CrossRef]

1997 (2)

1996 (1)

1994 (1)

K.-N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climate implications,” Atmos. Res. 31, 271–298 (1994).
[CrossRef]

1993 (2)

Y.-X. Hu, K. Stamnes, “An accurate parameterization of the radiative properties of water clouds suitable for use in climate models,” J. Clim. 6, 728–742 (1993).
[CrossRef]

L. Kou, D. Labrie, P. Chylek, “Refractive indices of water and ice in the 0.65- to 2.5-µm spectral range,” Appl. Opt. 32, 3531–3540 (1993).
[CrossRef] [PubMed]

1991 (1)

D. K. Perovich, J. W. Govoni, “Absorption coefficients of ice from 250 to 400 nm,” Geophys. Res. Lett. 18, 1233–1235 (1991).
[CrossRef]

1988 (1)

E. P. Zege, A. A. Kokhanovsky, “Integral characteristics of light scattering by large spherical particles,” Izv. Atmos. Ocean. Phys. 24, 508–512 (1988).

1984 (1)

1981 (1)

1980 (2)

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

1979 (1)

S. C. Colbeck, “Grain clusters in wet snow,” J. Colloid Interface Sci. 72, 371–384 (1979).
[CrossRef]

1974 (1)

C. F. Bohren, B. R. Barkstrom, “Theory of the optical properties of snow,” J. Geophys. Res. 79, 4527–4535 (1974).
[CrossRef]

1971 (1)

K. N. Liou, J. E. Hansen, “Intensity and polarization for single scattering by polydisperse spheres: a comparison of ray optics and Mie theory,” J. Atmos. Sci. 28, 995–1004 (1971).
[CrossRef]

Allison, I.

R. A. Massom, H. Eicken, C. Haas, M. O. Jeffries, M. R. Drinkwater, M. Sturm, A. P. Worby, X. Wu, V. I. Lytle, S. Ushio, K. Morris, P. A. Reid, S. G. Warren, I. Allison, “Snow on Antarctic sea ice,” Rev. Geophys. 39, 413–445 (2001).
[CrossRef]

R. A. Massom, V. I. Lytle, A. P. Worby, I. Allison, “Winter snow cover variability on East Antarctic sea ice,” J. Geophys. Res. 103, 24837–24855 (1998).
[CrossRef]

Bager, H.

H. Bager, Physics and Mechanics of Snow as a Material (U. S. Army Cold Region Research and Engineering Laboratory, Hanover, N.H., 1962).

Barkstrom, B. R.

C. F. Bohren, B. R. Barkstrom, “Theory of the optical properties of snow,” J. Geophys. Res. 79, 4527–4535 (1974).
[CrossRef]

Bohren, C. F.

C. F. Bohren, B. R. Barkstrom, “Theory of the optical properties of snow,” J. Geophys. Res. 79, 4527–4535 (1974).
[CrossRef]

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

Carlson, B. E.

Chen, S.-H.

Chylek, P.

Colbeck, S. C.

S. C. Colbeck, “Grain clusters in wet snow,” J. Colloid Interface Sci. 72, 371–384 (1979).
[CrossRef]

Davis, P. J.

P. J. Davis, P. Rabinowitz, Methods of Numerical Integration, 2nd ed. (Academic, Orlando, Fla., 1984), pp. 481–483.

Dozier, J.

A. W. Nolin, J. Dozier, “A hyperspectral method for remotely sensing the grain size of snow,” Remote Sens. Environ. 74, 207–216 (2000).
[CrossRef]

Drinkwater, M. R.

R. A. Massom, H. Eicken, C. Haas, M. O. Jeffries, M. R. Drinkwater, M. Sturm, A. P. Worby, X. Wu, V. I. Lytle, S. Ushio, K. Morris, P. A. Reid, S. G. Warren, I. Allison, “Snow on Antarctic sea ice,” Rev. Geophys. 39, 413–445 (2001).
[CrossRef]

R. A. Massom, M. R. Drinkwater, C. Haas, “Winter snow cover on sea ice in the Weddell Sea,” J. Geophys. Res. 102, 1101–1117 (1997).
[CrossRef]

Eicken, H.

R. A. Massom, H. Eicken, C. Haas, M. O. Jeffries, M. R. Drinkwater, M. Sturm, A. P. Worby, X. Wu, V. I. Lytle, S. Ushio, K. Morris, P. A. Reid, S. G. Warren, I. Allison, “Snow on Antarctic sea ice,” Rev. Geophys. 39, 413–445 (2001).
[CrossRef]

Eide, H. A.

H. A. Eide, J. J. Stamnes, K. Stamnes, F. M. Schulz, “New method for computing expansion coefficients for spheroidal functions,” J. Quant. Spectrosc. Radiat. Transfer 63, 191–203 (1999).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 350–354.

Glantschnig, W. J.

Govoni, J. W.

D. K. Perovich, J. W. Govoni, “Absorption coefficients of ice from 250 to 400 nm,” Geophys. Res. Lett. 18, 1233–1235 (1991).
[CrossRef]

Haas, C.

R. A. Massom, H. Eicken, C. Haas, M. O. Jeffries, M. R. Drinkwater, M. Sturm, A. P. Worby, X. Wu, V. I. Lytle, S. Ushio, K. Morris, P. A. Reid, S. G. Warren, I. Allison, “Snow on Antarctic sea ice,” Rev. Geophys. 39, 413–445 (2001).
[CrossRef]

C. Haas, “The seasonal cycle of ERS scatterometer signatures over perennial Antarctic sea ice and associated surface ice properties and processes,” Ann. Glaciol. 33, 69–73 (2001).
[CrossRef]

R. A. Massom, M. R. Drinkwater, C. Haas, “Winter snow cover on sea ice in the Weddell Sea,” J. Geophys. Res. 102, 1101–1117 (1997).
[CrossRef]

Hansen, J. E.

K. N. Liou, J. E. Hansen, “Intensity and polarization for single scattering by polydisperse spheres: a comparison of ray optics and Mie theory,” J. Atmos. Sci. 28, 995–1004 (1971).
[CrossRef]

Hecht, E.

E. Hecht, Optics, 3rd ed. (Addison-Wesley Longman, Reading, Mass., 1998).

Hu, Y.-X.

Y.-X. Hu, K. Stamnes, “An accurate parameterization of the radiative properties of water clouds suitable for use in climate models,” J. Clim. 6, 728–742 (1993).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Jeffries, M. O.

R. A. Massom, H. Eicken, C. Haas, M. O. Jeffries, M. R. Drinkwater, M. Sturm, A. P. Worby, X. Wu, V. I. Lytle, S. Ushio, K. Morris, P. A. Reid, S. G. Warren, I. Allison, “Snow on Antarctic sea ice,” Rev. Geophys. 39, 413–445 (2001).
[CrossRef]

K. Morris, M. O. Jeffries, “Seasonal contrasts in snow cover characteristics on Ross Sea ice floes,” Ann. Glaciol. 33, 61–68 (2001).
[CrossRef]

Kokhanovsky, A. A.

A. A. Kokhanovsky, T. Y. Nakajima, “The dependence of phase functions of large transparent particles on their refractive index and shape,” J. Phys. D 31, 1329–1335 (1998).
[CrossRef]

E. P. Zege, A. A. Kokhanovsky, “Integral characteristics of light scattering by large spherical particles,” Izv. Atmos. Ocean. Phys. 24, 508–512 (1988).

A. A. Kokhanovsky, Optics of Light Scattering Media: Problems and Solutions (Praxis, Chichester, UK, 1999).

T. Y. Nakajima, T. Nakajima, A. A. Kokhanovsky, “Radiative transfer through light-scattering media with nonspherical large particles: direct and inverse problems,” in Satellite Remote Sensing of Clouds and the Atmosphere II, J. D. Haigh, ed., Proc. SPIE3220, 2–12 (1998).
[CrossRef]

Kou, L.

Labrie, D.

Liou, K. N.

P. Yang, K. N. Liou, “Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35, 6568–6584 (1996).
[CrossRef] [PubMed]

K. N. Liou, J. E. Hansen, “Intensity and polarization for single scattering by polydisperse spheres: a comparison of ray optics and Mie theory,” J. Atmos. Sci. 28, 995–1004 (1971).
[CrossRef]

Liou, K.-N.

K.-N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climate implications,” Atmos. Res. 31, 271–298 (1994).
[CrossRef]

Lytle, V. I.

R. A. Massom, H. Eicken, C. Haas, M. O. Jeffries, M. R. Drinkwater, M. Sturm, A. P. Worby, X. Wu, V. I. Lytle, S. Ushio, K. Morris, P. A. Reid, S. G. Warren, I. Allison, “Snow on Antarctic sea ice,” Rev. Geophys. 39, 413–445 (2001).
[CrossRef]

R. A. Massom, V. I. Lytle, A. P. Worby, I. Allison, “Winter snow cover variability on East Antarctic sea ice,” J. Geophys. Res. 103, 24837–24855 (1998).
[CrossRef]

Macke, A.

Massom, R.

M. Sturm, K. Morris, R. Massom, “The winter snow cover of the West Antarctic pack ice: its spatial and temporal variability,” in Antarctic Sea Ice: Physical Processes, Interactions and Variability, M. O. Jeffries, ed., Vol. 74 of Antarctic Research Series (American Geophysical Union, Washington, D.C., 1998), pp. 1–18.

Massom, R. A.

R. A. Massom, H. Eicken, C. Haas, M. O. Jeffries, M. R. Drinkwater, M. Sturm, A. P. Worby, X. Wu, V. I. Lytle, S. Ushio, K. Morris, P. A. Reid, S. G. Warren, I. Allison, “Snow on Antarctic sea ice,” Rev. Geophys. 39, 413–445 (2001).
[CrossRef]

R. A. Massom, V. I. Lytle, A. P. Worby, I. Allison, “Winter snow cover variability on East Antarctic sea ice,” J. Geophys. Res. 103, 24837–24855 (1998).
[CrossRef]

R. A. Massom, M. R. Drinkwater, C. Haas, “Winter snow cover on sea ice in the Weddell Sea,” J. Geophys. Res. 102, 1101–1117 (1997).
[CrossRef]

Mishchenko, M. I.

Morris, K.

R. A. Massom, H. Eicken, C. Haas, M. O. Jeffries, M. R. Drinkwater, M. Sturm, A. P. Worby, X. Wu, V. I. Lytle, S. Ushio, K. Morris, P. A. Reid, S. G. Warren, I. Allison, “Snow on Antarctic sea ice,” Rev. Geophys. 39, 413–445 (2001).
[CrossRef]

K. Morris, M. O. Jeffries, “Seasonal contrasts in snow cover characteristics on Ross Sea ice floes,” Ann. Glaciol. 33, 61–68 (2001).
[CrossRef]

M. Sturm, K. Morris, R. Massom, “The winter snow cover of the West Antarctic pack ice: its spatial and temporal variability,” in Antarctic Sea Ice: Physical Processes, Interactions and Variability, M. O. Jeffries, ed., Vol. 74 of Antarctic Research Series (American Geophysical Union, Washington, D.C., 1998), pp. 1–18.

Nakajima, T.

T. Y. Nakajima, T. Nakajima, A. A. Kokhanovsky, “Radiative transfer through light-scattering media with nonspherical large particles: direct and inverse problems,” in Satellite Remote Sensing of Clouds and the Atmosphere II, J. D. Haigh, ed., Proc. SPIE3220, 2–12 (1998).
[CrossRef]

Nakajima, T. Y.

A. A. Kokhanovsky, T. Y. Nakajima, “The dependence of phase functions of large transparent particles on their refractive index and shape,” J. Phys. D 31, 1329–1335 (1998).
[CrossRef]

T. Y. Nakajima, T. Nakajima, A. A. Kokhanovsky, “Radiative transfer through light-scattering media with nonspherical large particles: direct and inverse problems,” in Satellite Remote Sensing of Clouds and the Atmosphere II, J. D. Haigh, ed., Proc. SPIE3220, 2–12 (1998).
[CrossRef]

Nolin, A. W.

A. W. Nolin, J. Dozier, “A hyperspectral method for remotely sensing the grain size of snow,” Remote Sens. Environ. 74, 207–216 (2000).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, London, 1992).
[CrossRef]

Perovich, D. K.

D. K. Perovich, J. W. Govoni, “Absorption coefficients of ice from 250 to 400 nm,” Geophys. Res. Lett. 18, 1233–1235 (1991).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 350–354.

Rabinowitz, P.

P. J. Davis, P. Rabinowitz, Methods of Numerical Integration, 2nd ed. (Academic, Orlando, Fla., 1984), pp. 481–483.

Reid, P. A.

R. A. Massom, H. Eicken, C. Haas, M. O. Jeffries, M. R. Drinkwater, M. Sturm, A. P. Worby, X. Wu, V. I. Lytle, S. Ushio, K. Morris, P. A. Reid, S. G. Warren, I. Allison, “Snow on Antarctic sea ice,” Rev. Geophys. 39, 413–445 (2001).
[CrossRef]

Schulz, F. M.

H. A. Eide, J. J. Stamnes, K. Stamnes, F. M. Schulz, “New method for computing expansion coefficients for spheroidal functions,” J. Quant. Spectrosc. Radiat. Transfer 63, 191–203 (1999).
[CrossRef]

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
[CrossRef]

Stamnes, J. J.

H. A. Eide, J. J. Stamnes, K. Stamnes, F. M. Schulz, “New method for computing expansion coefficients for spheroidal functions,” J. Quant. Spectrosc. Radiat. Transfer 63, 191–203 (1999).
[CrossRef]

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
[CrossRef]

Stamnes, K.

H. A. Eide, J. J. Stamnes, K. Stamnes, F. M. Schulz, “New method for computing expansion coefficients for spheroidal functions,” J. Quant. Spectrosc. Radiat. Transfer 63, 191–203 (1999).
[CrossRef]

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
[CrossRef]

Y.-X. Hu, K. Stamnes, “An accurate parameterization of the radiative properties of water clouds suitable for use in climate models,” J. Clim. 6, 728–742 (1993).
[CrossRef]

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, Cambridge, UK, 1999).
[CrossRef]

Sturm, M.

R. A. Massom, H. Eicken, C. Haas, M. O. Jeffries, M. R. Drinkwater, M. Sturm, A. P. Worby, X. Wu, V. I. Lytle, S. Ushio, K. Morris, P. A. Reid, S. G. Warren, I. Allison, “Snow on Antarctic sea ice,” Rev. Geophys. 39, 413–445 (2001).
[CrossRef]

M. Sturm, K. Morris, R. Massom, “The winter snow cover of the West Antarctic pack ice: its spatial and temporal variability,” in Antarctic Sea Ice: Physical Processes, Interactions and Variability, M. O. Jeffries, ed., Vol. 74 of Antarctic Research Series (American Geophysical Union, Washington, D.C., 1998), pp. 1–18.

Takano, Y.

K.-N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climate implications,” Atmos. Res. 31, 271–298 (1994).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 350–354.

Thomas, G. E.

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, Cambridge, UK, 1999).
[CrossRef]

Ushio, S.

R. A. Massom, H. Eicken, C. Haas, M. O. Jeffries, M. R. Drinkwater, M. Sturm, A. P. Worby, X. Wu, V. I. Lytle, S. Ushio, K. Morris, P. A. Reid, S. G. Warren, I. Allison, “Snow on Antarctic sea ice,” Rev. Geophys. 39, 413–445 (2001).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 350–354.

Warren, S. G.

R. A. Massom, H. Eicken, C. Haas, M. O. Jeffries, M. R. Drinkwater, M. Sturm, A. P. Worby, X. Wu, V. I. Lytle, S. Ushio, K. Morris, P. A. Reid, S. G. Warren, I. Allison, “Snow on Antarctic sea ice,” Rev. Geophys. 39, 413–445 (2001).
[CrossRef]

S. G. Warren, “Optical constants of ice from the ultraviolet to the microwave,” Appl. Opt. 23, 1206–1225 (1984). See Ref. 26 for the upgrade.

Wielaard, D. J.

Wiscombe, W. J.

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

W. J. Wiscombe, “Mie scattering calculations: advances in technique and fast, vector-speed computer codes,” (National Center for Atmospheric Research, Boulder, Colo., 1979).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

Worby, A. P.

R. A. Massom, H. Eicken, C. Haas, M. O. Jeffries, M. R. Drinkwater, M. Sturm, A. P. Worby, X. Wu, V. I. Lytle, S. Ushio, K. Morris, P. A. Reid, S. G. Warren, I. Allison, “Snow on Antarctic sea ice,” Rev. Geophys. 39, 413–445 (2001).
[CrossRef]

R. A. Massom, V. I. Lytle, A. P. Worby, I. Allison, “Winter snow cover variability on East Antarctic sea ice,” J. Geophys. Res. 103, 24837–24855 (1998).
[CrossRef]

Wu, X.

R. A. Massom, H. Eicken, C. Haas, M. O. Jeffries, M. R. Drinkwater, M. Sturm, A. P. Worby, X. Wu, V. I. Lytle, S. Ushio, K. Morris, P. A. Reid, S. G. Warren, I. Allison, “Snow on Antarctic sea ice,” Rev. Geophys. 39, 413–445 (2001).
[CrossRef]

Yang, P.

Zege, E. P.

E. P. Zege, A. A. Kokhanovsky, “Integral characteristics of light scattering by large spherical particles,” Izv. Atmos. Ocean. Phys. 24, 508–512 (1988).

Zhou, X.

X. Zhou, “Optical remote sensing of snow on sea ice: ground measurements, satellite data analysis, and radiative transfer modeling,” (Ph.D. thesis, University of Alaska, Fairbanks, Alaska, 2002).

Ann. Glaciol. (2)

C. Haas, “The seasonal cycle of ERS scatterometer signatures over perennial Antarctic sea ice and associated surface ice properties and processes,” Ann. Glaciol. 33, 69–73 (2001).
[CrossRef]

K. Morris, M. O. Jeffries, “Seasonal contrasts in snow cover characteristics on Ross Sea ice floes,” Ann. Glaciol. 33, 61–68 (2001).
[CrossRef]

Appl. Opt. (8)

Atmos. Res. (1)

K.-N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climate implications,” Atmos. Res. 31, 271–298 (1994).
[CrossRef]

Geophys. Res. Lett. (1)

D. K. Perovich, J. W. Govoni, “Absorption coefficients of ice from 250 to 400 nm,” Geophys. Res. Lett. 18, 1233–1235 (1991).
[CrossRef]

Izv. Atmos. Ocean. Phys. (1)

E. P. Zege, A. A. Kokhanovsky, “Integral characteristics of light scattering by large spherical particles,” Izv. Atmos. Ocean. Phys. 24, 508–512 (1988).

J. Atmos. Sci. (1)

K. N. Liou, J. E. Hansen, “Intensity and polarization for single scattering by polydisperse spheres: a comparison of ray optics and Mie theory,” J. Atmos. Sci. 28, 995–1004 (1971).
[CrossRef]

J. Clim. (1)

Y.-X. Hu, K. Stamnes, “An accurate parameterization of the radiative properties of water clouds suitable for use in climate models,” J. Clim. 6, 728–742 (1993).
[CrossRef]

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S. C. Colbeck, “Grain clusters in wet snow,” J. Colloid Interface Sci. 72, 371–384 (1979).
[CrossRef]

J. Geophys. Res. (3)

R. A. Massom, M. R. Drinkwater, C. Haas, “Winter snow cover on sea ice in the Weddell Sea,” J. Geophys. Res. 102, 1101–1117 (1997).
[CrossRef]

R. A. Massom, V. I. Lytle, A. P. Worby, I. Allison, “Winter snow cover variability on East Antarctic sea ice,” J. Geophys. Res. 103, 24837–24855 (1998).
[CrossRef]

C. F. Bohren, B. R. Barkstrom, “Theory of the optical properties of snow,” J. Geophys. Res. 79, 4527–4535 (1974).
[CrossRef]

J. Phys. D (1)

A. A. Kokhanovsky, T. Y. Nakajima, “The dependence of phase functions of large transparent particles on their refractive index and shape,” J. Phys. D 31, 1329–1335 (1998).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

H. A. Eide, J. J. Stamnes, K. Stamnes, F. M. Schulz, “New method for computing expansion coefficients for spheroidal functions,” J. Quant. Spectrosc. Radiat. Transfer 63, 191–203 (1999).
[CrossRef]

Phys. Rev. Lett. (1)

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

Remote Sens. Environ. (1)

A. W. Nolin, J. Dozier, “A hyperspectral method for remotely sensing the grain size of snow,” Remote Sens. Environ. 74, 207–216 (2000).
[CrossRef]

Rev. Geophys. (1)

R. A. Massom, H. Eicken, C. Haas, M. O. Jeffries, M. R. Drinkwater, M. Sturm, A. P. Worby, X. Wu, V. I. Lytle, S. Ushio, K. Morris, P. A. Reid, S. G. Warren, I. Allison, “Snow on Antarctic sea ice,” Rev. Geophys. 39, 413–445 (2001).
[CrossRef]

Other (15)

X. Zhou, “Optical remote sensing of snow on sea ice: ground measurements, satellite data analysis, and radiative transfer modeling,” (Ph.D. thesis, University of Alaska, Fairbanks, Alaska, 2002).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

H. Bager, Physics and Mechanics of Snow as a Material (U. S. Army Cold Region Research and Engineering Laboratory, Hanover, N.H., 1962).

M. Sturm, K. Morris, R. Massom, “The winter snow cover of the West Antarctic pack ice: its spatial and temporal variability,” in Antarctic Sea Ice: Physical Processes, Interactions and Variability, M. O. Jeffries, ed., Vol. 74 of Antarctic Research Series (American Geophysical Union, Washington, D.C., 1998), pp. 1–18.

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, London, 1992).
[CrossRef]

T. Y. Nakajima, T. Nakajima, A. A. Kokhanovsky, “Radiative transfer through light-scattering media with nonspherical large particles: direct and inverse problems,” in Satellite Remote Sensing of Clouds and the Atmosphere II, J. D. Haigh, ed., Proc. SPIE3220, 2–12 (1998).
[CrossRef]

W. J. Wiscombe, “Mie scattering calculations: advances in technique and fast, vector-speed computer codes,” (National Center for Atmospheric Research, Boulder, Colo., 1979).

E. Hecht, Optics, 3rd ed. (Addison-Wesley Longman, Reading, Mass., 1998).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

A. A. Kokhanovsky, Optics of Light Scattering Media: Problems and Solutions (Praxis, Chichester, UK, 1999).

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, Cambridge, UK, 1999).
[CrossRef]

P. J. Davis, P. Rabinowitz, Methods of Numerical Integration, 2nd ed. (Academic, Orlando, Fla., 1984), pp. 481–483.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 350–354.

The updated compilation of the ice optical constant was done by B. Gao, W. Wiscombe, S. Warren and is available by anonymous ftp to climate.gsfc.nasa.gov in the directory/pub/Wiscombe/Refrac_Index/ICE.

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Figures (10)

Fig. 1
Fig. 1

Ray-tracing diagram. Incident rays will propagate by separation of reflection and refraction. The energy contained in each ray decreases rapidly as it propagates. Rays such as A, B, and C, etc., emerging from the scatterer will experience the same process with other scatterers, resulting in multiple scattering. The weight of each arrowed line denotes the energy content contained in the ray.

Fig. 2
Fig. 2

(a) Maximum number N versus incidence angle for the truncation in the sum series of the calculation of absorption efficiency for an ice sphere with radius a = 1 cm and truncation tolerance ς = 1.0 × 10-16 and 1.0 × 10-4 for wavelength λ = 0.5 µm. (b) The maximum number N versus wavelength λ for the truncation in the sum series of the calculation of absorption efficiency of an ice sphere with radius a = 1 cm and truncation tolerance ς = 1.0 × 10-16 for incidence angle Θi = 60°.

Fig. 3
Fig. 3

(a) Absorption efficiency versus size parameter x = 2πa/λ for wavelength λ = 0.5 µm corresponding to refractive index m = 1.313 + i1.910 × 10-9. The absorption efficiency is calculated using Mie scattering theory (dashed curve) and GOM (solid curve). (b) Percentage difference Δabs = [(QabsMie - QabsGeo)/QabsMie]100% versus size parameter.

Fig. 4
Fig. 4

(a) Absorption efficiency versus size parameter x = 2πa/λ for wavelength λ = 2.0 µm corresponding to refractive index m = 1.274 + i1.640 × 10-3. Absorption efficiency is calculated by using the Mie scattering theory (dashed curve) and GOM (solid curve). (b) Percentage difference Δabs = [(QabsMie - QabsGeo)/QabsMie]100% versus size parameter.

Fig. 5
Fig. 5

Near-field scattering efficiency versus size parameter x = 2πa/λ for wavelength λ = 2.0 µm. The far-field scattering efficiency for λ = 2.0 µm is also included for comparison. The tolerance number ς = 10-8. The difference between the far-field and the near-field scattering efficiencies is due only to Fraunhofer diffraction.

Fig. 6
Fig. 6

Flow chart for the computational procedure adopted to compute the phase function for near-field scattering. T = true, F = false. The root finding is based on a combination of bracketing method and a hybrid algorithm based on the bisection and the Newton-Raphson methods. The truncation number is determined by Eq. (4).

Fig. 7
Fig. 7

Phase function for near-field scattering versus scattering angle Θ for a snow sphere of radius a = 1 cm for wavelength λ = 0.5 and 2.0 µm. Each peak in the visible curve corresponds to the position of a rainbow. The tolerance number ς = 10-8.

Fig. 8
Fig. 8

Comparison of the phase function calculated with GOMsphere with that from a Monte Carlo code21,23 for the nonabsorbing case of = 1.333 + i0.0. The tolerance number ς = 10-8. The phase function calculated by using N = 2 rather than by using the truncation formula [Eq. (4)] is also shown.

Fig. 9
Fig. 9

(a) Far-field scattering efficiency versus size parameter x = 2πa/λ for wavelength λ = 2.0 µm. In the inset the size parameter extends to 10,000. The difference between Mie and GOM calculations is very small except for small size parameters (<200). (b) The deviation of the scattering coefficient calculated by the GOMsphere code from that obtained by the MIE0 code. When x ≥ 110, the deviation is smaller than 5%. For both (a) and (b) the truncation tolerance number ς = 10-8.

Fig. 10
Fig. 10

Phase function for far-field scattering versus scattering angle Θ for a snow sphere of radius a = 1 cm for wavelength (a) λ = 0.5 µm and (b) λ = 2.0 µm. The truncation tolerance number ς = 10-8. For comparison, near-field results are also shown. Each peak in the visible curve corresponds to the position of a rainbow. Strong forward scattering due to Fraunhofer diffraction appears in the far-field scattering phase function.

Equations (42)

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Fλ=m2Z0 |E0|2 exp-βz,
Ep,rj=rpjEp,ij, Ep,tj=tpjEp,ij,
Ep,ij=Ep,i,j=1,Ep,ij=tpl=2jrplexp-βξj-1/2Ep,i=tp-rpj-2 exp-βξj-1/2Ep,i,j2,
rpj=rpΘi, Θt, mˆ=rp, tpj=tpΘi, Θt, mˆ=tpif j=1, andrpj=rpΘt, Θi, 1/mˆ, tpj=tpΘt, Θi, 1/mˆ,if j>1,
rΘi, Θt, mˆ=m2-m2cos Θi-u+i2mm cos Θi-vm2-m2cos Θi+u+i2mm cos Θi+v,
tΘi, Θt, mˆ=2m+imcos Θim2-m2cos Θi+u+i2mm cos Θi+v,
rΘi, Θt, mˆ=cos Θi-u-ivcos Θi+u+iv,
tΘi, Θt, mˆ=2 cos Θicos Θi+u+iv,
rpΘt, Θi, 1/mˆ=-rpΘi, Θt, mˆ, tpΘt, Θi, 1/mˆ=u+ivcos Θi tpΘi, Θt, mˆ,
u=22m2-m2-sin2 Θi2+4m2m21/2+m2-m2-sin2 Θi1/2, v=22m2-m2-sin2 Θi2+4m2m21/2-m2-m2-sin2 Θi1/2,
NΘi=βξ+2 ln|t|-2 ln R-ln ςβξ-ln R,
T=mum+vm2+vm-um21/2m2+m2cos Θi |t|2,
Wabs=2πa2F001 Tμij=2NR exp-βξj-2×1-exp-βξμidμi,
T=12T+T,
Tp=mum+vm2+vm-um21/2m2+m2cos Θi |tp|2,
R=12R+R,
ξ=|2a cos Θt|=2am2+m2um+vm2+vm-um21/2,
Cabs=WabsF0=2πa201 Tμij=2NR exp-βξj-2×1-exp-βξμidμi =2πa201 Tμi1-exp-βξ1-R exp-βξ μidμi,
Qabs=Cabsπa2.
Pτ; μ, ϕ; μ, ϕ=4πCscadτ; μ, ϕ; μ, ϕCsca=4πCscadτ, ΘCsca,
Csca=2π 0πdΘCscadτ, Θsin Θ,
Pτ; μ, ϕ; μ, ϕ=4πCscadτ, ΘCsca=2Cscadτ, Θ0πdΘCscadτ, Θsin Θ.
g=cos Θ=14π  Pτ; Θcos ΘdΩ=12-11 Pτ; Θcos Θd cos Θ.
Csca=j=1N WscajF0=Cdif+Cref+Ctra,
Qsca=Cscaπa2.
Cref=Wsca1F0=2πa201 Rμiμidμi,
Ctra=j=2N WscajF0=πa2j=2N01T2μiRj-2μi+T2μiRj-2μiexp-βξj-1μidμi.
Fpj, Θi, Θ=Cpdj, Θi, Θr2 F0=a2|pj|2Dj, Θi, ΘF0r2,
Dj, Θi, Θ=sin2Θi4j-1cos Θiu2+v21/2-1sin Θ, pj=rp,j=1u+ivcos Θitp2-rpj-2 exp-βξj-1/2,j2.
PΘ=2πCscaj,Θip Cpdj, Θi, Θ,
Csca=πa20πj,Θi|j|2+|j|2Dj, Θi, Θsin ΘdΘ.
Cext=Csca+Cabs.
CscadΘi, Θ=12j,Θip Cpdj, Θi, Θ+a2J12x sin Θsin2 Θ.
PΘi, Θ=2 j,Θip |pj|2Dj, Θi, Θ+4J12x sin Θsin2 Θ1+0πdΘ sin Θ j,Θippj2Dj, Θi, Θ.
Δabs=QabsMie-QabsGeoQabsMie 100%.
PΘ=2 j,Θip |pj|2Dj, Θi, Θ0πj,Θi|j|2+|j|2Dj, Θi, Θsin ΘdΘ,
g=0πj,Θip |pj|2Dj, Θi, Θcos Θ sin ΘdΘ0πj,Θi|j|2+|j|2Dj, Θi, Θsin ΘdΘ.
cos Θ=cos2j-1|Θt|-2Θi-j-2π.
sin Θi=2S1+S2, cos Θi=1-S21+S2,
fS=-1j-21+S221-6S2+S4cos2j-1|Θt|+4S1-S2sin2j-1|Θt|-cos Θ=0,
Θt=sin-11mˆ2S1+S2=-i lni 1mˆ2S1+S2+1-1mˆ2S1+S221/2.
QscaF=QscaN+1.

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